Inverse Problems of Mathematical Physics

01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. This book surveys basic features of the comparatively newly emerged theory of inverse problems for equations of mathematical physics, i.e. of the problems aimed at defining the coeficients of a differential equation through some functionals of its solution. This kind of problem arises in various fields of science when trying to describe internal characteristics of a substance where physico-chemical processes take place by the results of observing these processes within the available range of measurements. The book offers in-depth coverage of inverse problems for second-order equations and for hyperbolic systems of first-order equations, including the kinematic problem of seismology, the Lamb dynamic problem for equations of the theory of elasticity, and the problem of electrodynamics.

Table of Contents:
Foreword by V.G. Yakhno INTRODUCTION Inverse problem concept: examples of formulating inverse problems On correctness of direct and inverse problems of mathematical physics INVERSE PROBLEMS FOR THE OPERATOR #TEX2HTML_WRAP_INLINE3211# Problems with nonfocused initial data Some aspects associated with the inverse problem for the equation Problems with a focused source of disturbance Reducing the problem with a focused source of disturbance to a linear, integral equation: necessary and sufficient conditions for the inverse problem solvability Inverse problems for differential equations in a limited domain Relationship with the Sturm--Liouville problem One-dimensional inverse problems for second-order linear hyperbolic equations Problem of determining the operator in a second-order hyperbolic equation INVERSE KINEMATIC PROBLEMS IN SEISMOLOGY Iconical equations, rays, and fronts Boundary rays; waveguides: a sufficient condition for the absence of waveguides and boundary rays-ray regularity One-dimensional inverse kinematic problems Linearized three-dimensional inverse problems Nonlinear three-dimensional inverse problems Inverse problems using inner sources Inverse kinematic problems for an anisotropic medium SECOND-ORDER EQUATIONS OF A HYPERBOLIC TYPE AND RELATED INVERSE PROBLEMS Fundamental solution and its differential properties Ray formulation of inverse problems for the coefficients at minor derivatives Inverse dynamic problem: linearization method General scheme of studying inverse problems for hyperbolic type equations INVERSE PROBLEMS FOR FIRST-ORDER LINEAR HYPERBOLIC SYSTEMS Systems of equations with a single spatial variable Inverse problems using focused sources of wave generation Problem of determining the right-hand part of a hyperbolic system Lamb one-dimensional inverse problem Lamb three-dimensional inverse problem within a linear approach Inverse problem for a system of Maxwell equations INVERSE PROBLEMS FOR PARABOLIC AND ELLIPTICAL TYPE SECOND-ORDER EQUATIONS Problem of determining the density of heat sources Problem of determining diffusion coefficients Relations among inverse problems for parabolic, elliptical, and hyperbolic type equations On specific formulations of inverse problems where the coefficient to be determined is independent of one of the variables References Index